Theorem ltexp2d

Description: Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	resqcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	ltexp2d.2	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
	ltexp2d.3	⊢ ( 𝜑 → 𝑁 ∈ ℤ )
	ltexp2d.4	⊢ ( 𝜑 → 1 < 𝐴 )
Assertion	ltexp2d	⊢ ( 𝜑 → ( 𝑀 < 𝑁 ↔ ( 𝐴 ↑ 𝑀 ) < ( 𝐴 ↑ 𝑁 ) ) )

Proof

Step	Hyp	Ref	Expression
1		resqcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		ltexp2d.2	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
3		ltexp2d.3	⊢ ( 𝜑 → 𝑁 ∈ ℤ )
4		ltexp2d.4	⊢ ( 𝜑 → 1 < 𝐴 )
5		ltexp2	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ 1 < 𝐴 ) → ( 𝑀 < 𝑁 ↔ ( 𝐴 ↑ 𝑀 ) < ( 𝐴 ↑ 𝑁 ) ) )
6	1 2 3 4 5	syl31anc	⊢ ( 𝜑 → ( 𝑀 < 𝑁 ↔ ( 𝐴 ↑ 𝑀 ) < ( 𝐴 ↑ 𝑁 ) ) )